For the case of 2-edge-connectivity, the above means that we can
maintain the bridges of a decremental graph and we can also maintain
the connected components and their sizes without revealing what the current randomized certificate looks like. All this is needed for the unique
perfect matching algorithm of Gabow, Kaplan, and Tarhan [1]}. The
algorithm is an extremely simple recursion based on the fact that
a graph with a unique perfect matching has a bridge and all components have
even sizes. The algorithm first asks for a bridge \((u,v)\)  of some component. If there
is none, the is no unique matching. Otherwise we remove \((u, v)\)  and check the
sizes of the components of \(u\)  and \(v\) . If they are odd, \((u,v)\)  is in the unique matching, and we remove all other incident edges. Otherwise \((u,v)\)  is not in the
unique matching. The important thing here is that the bridges do not tell us anything about our 2-certificate of the 2-edge-connected components.
